some results on a transformation of copulas and quasi-copulas

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Information Sciences xxx (2013) xxx–xxx

INS 10304 No. of Pages 7, Model 3G

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Contents lists available at ScienceDirect

Information Sciences

journal homepage: www.elsevier .com/locate / ins

Some results on a transformation of copulas and quasi-copulas

0020-0255/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.ins.2013.09.023

⇑ Corresponding author. Tel.: +98 3518250309.E-mail addresses: [email protected] (A. Dolati), [email protected] (M. Úbeda-Flores), [email protected] (M. Úbeda-Flores).

Please cite this article in press as: A. Dolati et al., Some results on a transformation of copulas and quasi-copulas, Inform. Sci. (2013)dx.doi.org/10.1016/j.ins.2013.09.023

Ali Dolati a,⇑, Soheyla Mohseni a, Manuel Úbeda-Flores b

a Department of Statistics, Yazd University, Yazd 89195-741, Iranb Departamento de Matemáticas, Universidad de Almería, Carretera de Sacramento s/n, 04120 La Cañada de San Urbano, Almería, Spain

a r t i c l e i n f o a b s t r a c t

26272829

Article history:Received 15 March 2013Received in revised form 20 July 2013Accepted 6 September 2013Available online xxxx

Keywords:CopulaDependence propertiesMarshal–Olkin copulaMeasures of associationQuasi-copulaTail dependence

This paper provides some results including invariance properties, dependence measures,convexity properties and tail dependence on a transformation of copulas andquasi-copulas.

� 2013 Elsevier Inc. All rights reserved.

30

1. Introduction

A (bivariate) copula is a function C: [0,1]2 ? [0,1] such that (C1) C(t,0) = C(0, t) = 0 and C(t,1) = C(1, t) = t for all t 2 [0,1], and(C2) VC([u1,u2] � [v1,v2]) = C(u2,v2) � C(u2,v1) � C(u1,v2) + C(u1,v1) P 0 for all u1, u2, v1, v2 in [0,1] such that u1 6 u2 and v1 6 v2.Copulas have proved to be a useful tool in the construction of multivariate distribution functions. In fact, in view of Sklar’s The-orem [24], the joint distribution H of a pair of random variables—defined on a common probability space ðX;PÞ—and the cor-responding marginal distributions F and G are linked by a copula C in the following manner: H(x,y) = C(F(x),G(y)) for all x,y in[�1,1]. For a complete review of this concept and some of its applications see [3,14,20]. Let P denote the copula forindependent random variables, i.e., P(u,v) = uv for all (u,v) in [0,1]2. Every copula C satisfies the following inequalities:

maxðuþ v � 1;0Þ ¼Wðu; vÞ 6 Cðu; vÞ 6 Mðu;vÞ ¼minðu;vÞ; 8ðu;vÞ 2 ½0;1�2;
where M and W are themselves copulas. Various procedures to construct copulas have been proposed in the literature (e.g.,see [15,20]). Recently, some authors provided construction methods from the class of copulas to itself, or from a moregeneral class of functions to another (e.g., see [2,6,8,16,19]).
For a given copula D and a, b 2 [0,1], consider the function Ca,b[D] defined by

Ca;b½D�ðu; vÞ ¼ u1�av1�bDðua;vbÞ; ð1Þ

for every (u,v) 2 [0,1]2. This structure is, in fact, a copula and first appeared in [12,17] as a mechanism for generatingasymmetric copulas. Indeed, if the pair (U1,U2) and (V1,V2) are two independent vectors of uniform (0,1) random variableswith associated copulas D and P, respectively, then Ca,b[D] is the joint distribution of the pair (U,V) defined by

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http://www.sciencedirect.com/science/journal/00200255

http://www.elsevier.com/locate/ins



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dx.doi

U ¼maxfU1=a1 ;V1=ð1�aÞ

1 g; V ¼maxfU1=b2 ;V1=ð1�bÞ

2 g:

A multivariate generalization of (1) is given in [18]. Observe that C0,0[D] = P, C1,1[D] = D and

Ca;b½M�ðu; vÞ ¼minðu1�av; uv1�bÞ; ð2Þ

is the Marshal–Olkin (which we denoted by MO) family of copulas [20]. To the best of our knowledge properties of the family(1) had not been thoroughly investigated. The aim of this paper is to provide some results for this family, including invari-ance properties, dependence measures, convexity properties and tail dependence (Section 2). In Section 3 we study the casein which D is a proper quasi-copula.

2. Properties

In this section we provide some results including invariance properties, dependence measures, convexity properties andtail dependence on the family (1).

2.1. Invariance

We say that a given copula D is Ca,b invariant under (1) if Ca,b[D] = D for every a, b 2 [0,1]. First note that the transforma-tion (1) is ‘‘unique’’, in the sense that if D1 and D2 are two copulas such that Ca,b[D1] = Ca,b[D2] for every a, b 2 [0,1], thenD1 = D2. Although we do not find a pattern to describe the class of Ca,b invariant copulas, we note that, for example, theGumbel–Barnett family of copulas [20], defined by

Dhðu;vÞ ¼ uve�h lnðuÞ lnðvÞ; 0 6 h 6 1;

satisfies Ca,b[Dh] = Dabh. The following example shows the invariance property of (1) in the class of extreme value copulas.

Example 1. Consider the family of extreme value copulas [22]

DAðu;vÞ ¼ exp lnðuvÞA lnðuÞlnðuvÞ

� �� ;

with the dependence function A: [0,1] ? [0,1/2], satisfying A(0) = A(1) = 1 and max (t,1 � t) 6 A(t) 6 1. Under the construc-tion (1) we see that Ca;b½DA� ¼ DAa;b , where

Aa;bðtÞ ¼ ð1� aÞt þ ð1� bÞð1� tÞ þ ðat þ bð1� tÞÞA atat þ bð1� tÞ

� �:

We note that the MO family of copulas (2) with the parameters k and c, is a member of the extreme value class of copulaswith the dependence function A(t) = 1 �min (kt,c(1 � t)) and the resulting copula under (1) is a gain a MO copula with thenew parameters ak and bc.

We also have the following result, whose proof is immediate and we omit it.

Proposition 1. For each copula D and a1, b1, a2, b2 2 [0,1], the copula given by (1) satisfies the stability propertyCa2 ;b2

Ca1 ;b1½D�

� �¼ Ca1a2 ;b1b2

½D�.

Example 2. Consider the FGM family of copulas, defined by Dh(u,v) = uv[1 + h(1 � u)(1 � v)] for all (u,v) 2 [0,1]2, withh 2 [�1,1]. Then we have that

Ca;b½Dh�ðu;vÞ ¼ uv 1þ hð1� uaÞð1� vbÞ� �

; ðu;vÞ 2 ½0;1�2; ð3Þ

which is an extension of the FGM family of copulas studied in [13]. The resulting family (3) is stable under (1).For any convex linear combination of copulas (which is again a copula [20]) the following result is immediate.

Proposition 2. Let Dk(u,v) = kD1(u,v) + (1 � k)D2(u,v), k 2 [0,1], be a linear convex combination of two copulas D1 and D2. Then

Ca;b½Dk�ðu;vÞ ¼ kCa;b½D1�ðu;vÞ þ ð1� kÞCa;b½D2�ðu; vÞ:
The following result shows the invariance of the copulas M, P and W.
Proposition 3. Given a copula D, and the transformation defined by (1) for a, b 2 (0,1], we have:

(i) Ca,b[D] = M if, and only if, D = M and a = b = 1.(ii) Ca,b[D] = W if, and only if, D = W and a = b = 1.(iii) Ca,b[D] = P if, and only if, D = P.

cite this article in press as: A. Dolati et al., Some results on a transformation of copulas and quasi-copulas, Inform. Sci. (2013), http://.org/10.1016/j.ins.2013.09.023



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Proof.

(i) Assume Ca,b[D] = M. Then we have u1�av1�bD(ua,vb) = M(u,v) for all (u,v) 2 [0,1]2. Suppose 0 < u 6 v < 1. Thenu1�av1�bD(ua,vb) = u, i.e., D(ua,vb) = uavb�1. Since each copula D satisfies D 6M, we have D(ua,vb) 6 ua, whenceuavb�1

6 ua, i.e., vb�16 1 for all v 2 (0,1); which is only true when b = 1. For the case 0 < v 6 u < 1 the proof is similar

and leads to a = 1. In this case, we obtain D = M. Conversely, the proof is obvious.(ii) Suppose Ca,b[D] = W. Then we have u1�av1�bD(ua,vb) = W(u,v) for all (u,v) 2 [0,1]2. If (u,v) 2 (0,1)2 are such that

u + v � 1 6 0, then u1�av1�bD(ua,vb) = 0, i.e., D(ua,vb) = 0 for all (u,v) 2 (0,1)2; but this implies a = b = 1, since ua > uand vb > v for all a, b 2 (0,1) and there exist (u,v) for which ua + vb � 1 > 0. Conversely, the result is trivial.

(iii) If Ca,b[D] = P, then we have u1�av1�bD(ua,vb) = uv, i.e., D(ua,vb) = uavb, which implies D = P. Conversely, the result isobvious and the proof is done. h

A copula D is said to be symmetric if D(u,v) = D(v,u) for all u, v 2 [0,1]. Note that, in general, the copula (1) is asymmetric,but if D is symmetric and a = b, then Ca,b[D] is symmetric as well.

2.2. Dependence concepts and ordering

Let C be the class of all bivariate copulas and let Ci; i ¼ 1; 2; 3; 4 be the subclasses defined by

Pleasedx.doi

C1 ¼ D 2 C : Dðu;vÞP uvf g;

C2 ¼ D 2 C :Dðu0;vÞ

u06

Dðu;vÞu

; 8 0 < u < u0; 8 0 6 v 6 1�

;

C3 ¼ D 2 C :Dðu;v 0Þ

v 0 6Dðu;vÞ

v ; 8 0 < v < v 0; 8 0 6 u 6 1�

;

C4 ¼ D 2 C : D is TP2f g

(we recall that a function A : R2 ! Rþ is TP2 if A(x0,y0)A(x00,y00) P A(x0,y00) A(x00,y0) with x0 < x00 and y0 < y00).Let (X,Y) be a pair of continuous random variables whose copula is D. Then (X,Y) is said to be positive quadrant dependent

(PQD) if D 2 C1. The random variable Y is said to be left tail decreasing in X (LTD (YjX)) if D 2 C2. If D 2 C3, then LTD (XjY). Therandom vector (X,Y) is left corner set decreasing (LCSD) if D 2 C3 (see [20] for a complete study).

Proposition 4. For each i = 1, 2, 3, 4, if D 2 Ci, then also Ca;b½D� 2 Ci for all a, b 2 [0,1].

Proof. For i = 1, the proof is immediate. For i = 2 (and similarly for i = 3), we note that for 0 < u < u0 6 1 and for all v 2 [0,1],the inequality Ca,b[D](u0,v)/u0 6 Ca,b[D](u,v)/u amounts to D(u0a,vb)/u0a 6 D(ua,vb)/ua, which holds for all D 2 C2 and a,b 2 [0,1]. For i = 4, if D 2 C3, then for all 0 6 u 6 u0 6 1, 0 6 v 6 v 0 6 1, we have u1�av1�bu01�av 01�b = u1�av 01�bu01�av1�b andD(ua,vb)D(u0a,v 0b) P D(ua,v 0b)D(u0a,vb), for all a, b 2 [0,1]. From these relations it follows

Ca;b½D�ðu; vÞCa;b½D�ðu0;v 0ÞP Ca;b½D�ðu;v 0ÞCa;b½D�ðu0;vÞ;

completing the proof. h

If C1 and C2 are two copulas, we say that C2 is more concordant than C1 (written C1 �c C2) if C1(u,v) 6 C2(u,v) for all(u,v) 2 [0,1]2. A totally ordered parametric family {Ck} of copulas is positively ordered if Ck1�cCk2 whenever k1 6 k2 [15,20].For the family (1) of copulas we have the following results.

Proposition 5. Let D be two copulas such that D1 �c D2. Then we have Ca,b[D1] �c Ca,b[D2] for every a, b 2 [0,1].

Proposition 6. Given a copula D, the parametric family {Ca,b[D]} of copulas is positively ordered if D 2 C2 \ C3.

Proof. Let 0 < a1 6 a2 6 1 and b1 6 b2. Then, for a given copula D and all u, v 2 [0,1], Ca1 ;b1 ½D�ðu;vÞ 6 Ca2 ;b2 ½D�, if, and only if,

Cðua1 ;vb1 Þua1 vb1

6Cðua2 ;vb2 Þ

ua2 vb2:

Setting uai ¼ ti and vbi ¼ si; i ¼ 1; 2, we get t1 P t2, s1 P s2 and above inequality amounts to D(t1,s1)/t1s1 6 D(t2,s2)/t2s2, forall t1 > t2 and s1 > s2. But the later holds if, and only if, D 2 C2 and D 2 C3. h

2.3. Measures of association

The population version of three of the most common nonparametric measures of association (in fact, measures of concor-dance, since they satisfy a set of axioms due to M. Scarsini [23]) between the components of a continuous random pair (X,Y)




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are Kendall’s tau (s), Spearman’s rho (q), and the medial correlation coefficient or Blomqvist’s beta (b). These measures dependonly on the copula D of the pair (X,Y), and are given by

Pleasedx.doi

sðDÞ ¼ 4Z 1

0

Z 1

0Dðx; yÞdDðx; yÞ � 1; ð4Þ

qðDÞ ¼ 12Z 1

0

Z 1

0Dðx; yÞ � xyð Þdxdy; ð5Þ

bðDÞ ¼ 4D12;12

� �� 1; ð6Þ

respectively (for a complete study, see [20] and the references therein). The direct computation of these measures in themodel given by (1) does not provide too much information; however, we have the following result, in which we find boundson these measures.

Proposition 7. For a given copula D and for each a 2 (0,1], the measures of association s, q and b associated with the family ofcopulas (1) satisfy the following inequalities:

2að4� bÞð2� aÞð2� bÞBeta

2a;2b

� �� 2abð2� aÞð2� bÞ 6 sðCa½D�Þ 6

aba� abþ b

;

12ð2� aÞð2� bÞBeta

2a;2b

� �� 3abð2� aÞð2� bÞ 6 qðCa½D�Þ 6

3ab2a� abþ 2b

;

maxð2a þ 2b � 2aþb; 0Þ � 1 6 bðCa½D�Þ 6 minð2a;2bÞ � 1;

where Betaða; bÞ ¼R 1

0 ta�1ð1� tÞb�1dt.

Proof. First of all, we recall that given two copulas C1 and C2 such that C1 �c C2, then we have s(C1) 6 s(C2), q(C1) 6 s(C2), andb(C1) 6 b(C2) [20]. Since any copula D satisfies W �c D �c M, from Proposition 5 we have Ca,b[W] �c Ca,b[D] �c Ca,b[M]. Thus,the upper bounds for these measures are equal to those of the MO family of copulas given by (2). By using the expressions in(4)–(6) to the copula Ca,b[W](u,v) = u1�av1�b max{ua + vb � 1,0} after some elementary (but tedious) algebra, we obtain thecorresponding lower bounds. h

2.4. Tail dependence

A reason for adding new parameters to a given copula is to produce families that exhibit some more flexible properties. Inparticular, copulas with different tail behavior are often useful to build models for estimating the extreme and risky events[15]. For a given copula D, the lower tail dependence coefficient is defined by

kLðDÞ ¼ limu!0þ

Dðu;uÞu

; ð7Þ

and the upper tail dependence coefficient as

kUðDÞ ¼ 2� limu!1�

1� Dðu;uÞ1� u

; ð8Þ

(see [15,20]).The next result shows how the proposed model (1) may modify the tail behavior of a given copula D, as measured by its

tail dependence coefficients.

Proposition 8. For a given copula D, we have kL(Ca,b[D]) = 0 for a, b 2 [0,1), kL(C1,1[D]) = kL(D) and kU(Ca,b[D]) = a + b �max(a,b)(2 � kU(D)) for every a, b 2 [0,1].

Proof. Since trivially kL(C1,1[D]) = kL(D), we first assume a, b 2 [0,1). By taking into account (7), the lower tail dependencecoefficient of Ca,b[D] can be expressed as

kLðCa;b½D�Þ ¼ limu!0þ

u2�a�bDðua; ubÞu

¼ limu!0þ

u1�b � limu!0þ

D ua;ubð Þua :

Since for a 6 b (resp, a P b), we have D(ua,ub)/ua6 D(ua,ua)/ua (resp, D(ua,ub)/ua P D(ua,ua)/ua), above expressions lead to

kL(Ca,b[D]) = 0 � kL(D) = 0. Now, for a, b 2 [0,1], by taking into account (8), the upper tail dependence coefficient of Ca,b[D] canbe expressed as




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dx.doi

kUðCa;b½D�Þ ¼ 2� limu!1�

1� u2�a�bDðua;ubÞ1� u

¼ 2� limu!1�

1� u2�a�b

1� uþ 1� Dðua;ubÞ

1� ub

� �u2�a�bð1� ubÞ

1� u

� �

¼ aþ b� b limu!1�

1� Dðua; ubÞ1� ub

;

or equivalently

kUðCa;b½D�Þ ¼ aþ b� a limu!1�

1� Dðua; ubÞ1� ua :

From the above expressions, it follows that kU(Ca,b[D]) = a + b �max (a,b)(2 � kU(D)), which completes the proof.

Remark 1. As a consequence of Proposition 8, if the copula D has no upper tail dependence, then the copula Ca,b[D] does.Moreover, even if D has a lower tail dependence, the lower tail dependence of Ca,b[D] is zero. We also note that for eachopula D, the upper tail dependence of the copula Ca,b[D] is smaller than those of the MO family of copulas, i.e., min (a,b).

Example 3. Let D be an Archimedean copula with generator u. It is known that

kUðDÞ ¼ 2� limt!0þ

1�u½�1�ð2tÞ1�u½�1�ðtÞ

[20]. Then we have

kUðCa;a½D�Þ ¼ a 2� limt!0þ

1�u½�1�ð2tÞ1�u½�1�ðtÞ

� �:

In particular, if D is an Archimedean copula whose generator is given by u(t) = eh/(t�1) for t 2 [0,1], with h 2 [2,1), then wehave kU(D) = 1, and hence kU(Ca,a[D]) = a.

2.5. Convexity properties

In this subsection we study some properties of convexity (concavity) for the copulas given by (1) when a = b. A copula C isSchur-concave if

Cðu; vÞ 6 Cðkuþ ð1� kÞv; kv þ ð1� kÞuÞ ð9Þ

for all u, v 2 (0,1) and k 2 [0,1] [20]. The following result shows that Schur-concavity of a given copula D is preserved underthe construction (1) (the class of Schur-concave copulas will be denoted by Csc).

Proposition 9. Let D be a Schur-concave copula. Then the generated copulas Ca,a[D] defined by (1) are Schur-concave as well.

Proof. First, we note that for all u, v 2 [0,1] and k 2 [0,1], [ku + (1 � k)v][kv + (1 � k)u] P uv. Since h(t) = ta is a concavefunction in t for all a 2 [0,1], then we have

kua þ ð1� kÞva6 kuþ ð1� kÞvð Þa

for all u, v 2 [0,1] and for all k, a 2 [0,1]. Moreover, since D 2 Csc and D(u,v) is non-decreasing in each variables, then

D ua; vað Þ 6 D kua þ ð1� kÞva; kva þ ð1� kÞuað Þ 6 D ðkuþ ð1� kÞvÞa; ðkv þ ð1� kÞuÞa �

;

and hence

ðuvÞ1�aDðua;vaÞ 6 ½kuþ ð1� kÞv �½kv þ ð1� kÞu�ð Þ1�aDððkuþ ð1� kÞvÞa; ðkv þ ð1� kÞuÞaÞ;

that is,

Ca;a½D�ðu;vÞ 6 Ca;a½D�ðkuþ ð1� kÞv; ð1� kÞuþ kvÞ;

as desired. h

A copula C is said to be quasi-concave if for all u, v, u0, v 0 2 [0,1] and all k 2 [0,1],

Cðkuþ ð1� kÞv; ku0 þ ð1� kÞv 0ÞP minfCðu; u0Þ; Cðv ;v 0Þg

(see [20] for details). The following result—whose proof is simple—shows that quasi-concavity of a given copula D is pre-served under the construction (1).




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Proposition 10. If D is quasi-concave, then the generated copula Ca,a[D] given by (1) is quasi-concave as well.A copula D is called supermigrative if it is symmetric and satisfies the inequality

Pleasedx.doi

Dðku; vÞP Dðu; kvÞ; ð10Þ

for all k 2 [0,1] and for all u, v 2 [0,1] such that v 6 u (see [7] for details). The class of supermigrative copulas will be denotedby Cþsm. The class of submigrative copulas, denoted by C�sm, is obtained by reversing the sense of the above inequality.

Proposition 11. Let D be a symmetric copula belongs to Cþsm (respectively, C�sm). Then the copula Ca;a½D� 2 Cþsm (respectively, C�sm).

Proof. Applying (10)—the case for submigrativity is similar, we have that Ca,a[D] is supermigrative if, and only if,

DfðkuÞa;vagP Dfua; ðkvÞag;

for all a, k 2 [0,1] and 0 6 v 6 u 6 1. Setting k0 = ka, t = ua and s = va, we get k0 2 [0,1] and 0 < s 6 t 6 1, and the above inequal-ity amounts to D(k0t,s) P D(t,k0s), which is D 2 Cþsm. h

3. A transformation of quasi-copulas

In the structure given by (1) we have assumed that D is a copula. We now wonder about the result if we suppose that D isa quasi-copula, a concept related to that of a copula. The notion of (bivariate) quasi-copula was introduced in [1] in order toshow that a certain class of operations on univariate distribution functions is not derivable from corresponding operations onrandom variables defined on the same probability space. A quasi-copula is a function Q: [0,1]2 ? [0,1] that satisfies theboundary conditions (C1), but instead of (C2), the weaker conditions [11]:

(Q1) Q is increasing in each variable; and(Q2) Q is 1-Lipschitz, i.e., for all u1, v1, u2, v2 2 [0,1] it holds that jQ(u1,v1) � Q(u2,v2)j 6 ju1 � u2j + jv1 � v2j.

Another characterization of a quasi-copula—which will be useful for our purposes—is given in the following result [11].

Lemma 12. A function Q: [0,1]2 ? [0,1] is a quasi-copula if and only if it satisfies condition (C1) and VQ(R) P 0 for every rectangleR = [u,u0] � [v,v 0] whenever 0 6 u 6 u0 6 1, 0 6 v 6 v 0 6 1 and at least one of u, u0, v or v 0 is either equal to 0 or to 1.

While every copula is a quasi-copula, there exist proper quasi-copulas, i.e., quasi-copulas that are not copulas (similaritiesand differences between copulas and proper quasi-copulas can be found, e.g., in [10,21]). However, quasi-copulas are alsobounded by the copulas W and M.

Theorem 13. Let D be a proper quasi-copula and 0 6 a 6 1. Then the function Ca,a[D] given by (1) is a proper quasi-copula.

Proof. In order to prove that (1) is a quasi-copula, we only need to check conditions (Q1) and (Q2). Let u, u0, v 2 [0,1] suchthat u < u0 (the proof for the second component is similar). Then we have Ca,a[D](u0,v) � Ca,a[D](u,v) = (u0v)1�aD((u0)a,va) �(uv)1�aD(ua,va). Since D is increasing in each variable and 1-Lipschitz, then

Ca;a½D�ðu0; vÞ � Ca;a½D�ðu;vÞP ðuvÞ1�a D ðu0Þa;va �� D ua; vað Þ

� �P 0;

and

Ca;a½D�ðu0; vÞ � Ca;a½D�ðu;vÞ ¼ ðu0vÞ1�aD ðu0Þa; va �� ðuvÞ1�aD ua;vað Þ þ ðu0vÞ1�aD ua; vað Þ � ðu0vÞ1�aD ua;vað Þ

6 ðu0vÞ1�a D ðu0Þa;va �� D ua;vað Þ

� �þ u0vð Þ1�a � ðuvÞ1�ah i

D ua;vað Þ

6 ðu0vÞ1�a ðu0Þa � ua� �þ ðu0vÞ1�a � ðuvÞ1�ah i

ua6 v1�aðu0 � uÞ 6 u0 � u;

i.e., conditions (Q1) and (Q2) are satisfied. Finally, since D is a proper quasi-copula, let R = [u1,u2] � [v1,v2] be a rectangle in[0,1]2 such that VD(R) < 0 (note that, in view of Lemma 12, we have 0 < u1 6 u2 < 1 and 0 < v1 6 v2 < 1). Then

VCa ½D�ðRÞ 6 u1�a1 ðv1�a

2 Dðua2; v

a2Þ � v1�a

1 Dðua2; v

a1Þ � v1�a

2 Dðua1;v

a2Þ � v1�a

1 Dðua1;v

a1Þ 6 u1v1ð Þ1�aVDðRÞ < 0;

i.e., Ca,a[D] is a proper quasi-copula, and this completes the proof.

4. Discussion

In this paper, we studied different properties of a transformation of copulas and quasi-copulas. Other transformations ofcopulas into copulas can be defined. For example, for two copulas A and B and functions f, g: [0,1] ? [0,1], let




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Cf ;g ½A;B�ðu; vÞ ¼ Aðf ðuÞ; gðvÞÞB uf ðuÞ ;

vgðvÞ

� �; ð11Þ

for every (u,v) 2 [0,1]2. According to the Corollary 3 in [9] this structure is always a copula, when the functions tf ðtÞ and t

gðtÞ areincreasing on [0,1]. Observe that letting A = P, fa(t) = t1�a and g(t) = t1�b with a, b 2 [0,1] we obtain the class of copulas de-fined by (1). In a future work we will thoroughly investigate different properties of the class of copulas defined by (11).

5. Uncited references

[4,5].

Acknowledgements

The authors thank Associate Editor and two anonymous referees for some enlightening remarks on an earlier version ofthis paper. The third author thanks the support by the Ministerio de Ciencia e Innovacion (Spain) and FEDER, under researchproject MTM2009-08724 and the Consejeria de Educacion y Ciencia of the Junta de Andalucia (Spain).

References

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some results on a transformation of copulas and quasi-copulas

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